# Order of reduction of infinite order rational point on an Elliptic Curve

Let $E/$ℚ be an elliptic curve and $P$ ∈ $E($ℚ$)$ a rational point of infinite order. Does the reduction of $P$ mod $p$ generate a maximal cyclic subgroup of $E(\mathbb{F}$$p$$)$ for almost all primes $p$?

• No. The primes that split in the field $\mathbb{Q}(Q)$, where $2Q=P$ are counterexamples. One expects a positive density of such primes, but not density one, if $P$ is not divisible over the rationals. – Felipe Voloch Sep 25 '15 at 0:36
• Already in $\mathbb{G}_m$ the answer is clearly negative, for the same reason that Felipe gave you. The statement on positive density can be proved conditionally on GRH. One can in fact say a lot more in the elliptic case: the group $E(\mathbb{F}_p)$ is the cyclic group $\langle P \mod{p} \rangle$ for a definite positive density of primes $p$, unless there is an obvious global obstruction ($P$ being divisible over $\mathbb{Q}$ or the rational torsion being non-cyclic). Those are elliptic variants of Artin's conjecture; Alina Cojocaru has obtained various interesting results in this direction. – Vesselin Dimitrov Sep 25 '15 at 0:52
• Thanks for good comments. I'll write Felipe Voloch's as an answer. – David Lampert Sep 25 '15 at 13:43

No. There is a positive density of primes that split in ℚ$(Q, E[2])$ (where $2Q=P$) and excluding the finitely many primes for which reduction of $E[2]$ isn't injective. For such primes any maximal cyclic subgroup of $E(\mathbb{F}$$p$$)$ has even order so reduction of $P$ can't be a generator (since reduction of $P$ is 2-divisible).
However, it is true that $$f_p=\text{order of \tilde P in \tilde E(\mathbb F_p)}$$ cannot be "too small, too often." For example, for every $\epsilon>0$, the series $$\sum_{p~\text{prime}} \frac{\log p}{p\cdot f_p^\epsilon}$$ converges. (More precisely, the series is ${}\le 3\epsilon^{-1}+O(1)$ as $\epsilon\to0$.)
• @DavidLampert: I believe it is also possible to prove that $f_P > p^{1/3}$ for almost all $p$, though I don't know if this has been written up anywhere. Up to an $\epsilon$ in the exponent, this appears in C. Matthews, Counting points modulo $p$ in some finitely generated subgroups of algebraic groups;this is a bit stronger than Theorem 4(b) (with $r = 1$) of the quoted paper. Improving the $1/3$ exponent is an open problem as far as I am aware, though on GRH it is possible to raise it all the way up to $1-\epsilon$. – Vesselin Dimitrov Nov 13 '15 at 18:40